Levene Model Research Articles

Evolutionarily stable mating decisions for sequentially searching females and the stability of reproductive isolation by assortative mating.

We consider mating strategies for females who search for males sequentially during a season of limited length. We show that the best strategy rejects a given male type if encountered before a time-threshold but accepts him after. For frequency-independent benefits, we obtain the optimal time-thresholds explicitly for both discrete and continuous distributions of males, and allow for mistakes being made in assessing the correct male type. When the benefits are indirect (genes for the offspring) and the population is under frequency-dependent ecological selection, the benefits depend on the mating strategy of other females as well. This case is particularly relevant to speciation models that seek to explore the stability of reproductive isolation by assortative mating under frequency-dependent ecological selection. We show that the indirect benefits are to be quantified by the reproductive values of couples, and describe how the evolutionarily stable time-thresholds can be found. We conclude with an example based on the Levene model, in which we analyze the evolutionarily stable assortative mating strategies and the strength of reproductive isolation provided by them.

Equilibrium properties of a multi-locus, haploid-selection, symmetric-viability model

Under haploid selection, a multi-locus, diallelic, two-niche Levene (1953) model is studied. Viability coefficients with symmetrically opposing directional selection in each niche are assumed, and with a further simplification that the most and least favored haplotype in each niche shares no alleles in common, and that the selection coefficients monotonically increase or decrease with the number of alleles shared. This model always admits a fully polymorphic symmetric equilibrium, which may or may not be stable.We show that a stable symmetric equilibrium can become unstable via either a supercritical or subcritical pitchfork bifurcation. In the supercritical bifurcation, the symmetric equilibrium bifurcates to a pair of stable fully polymorphic asymmetric equilibria; in the subcritical bifurcation, the symmetric equilibrium bifurcates to a pair of unstable fully polymorphic asymmetric equilibria, which then connect to either another pair of stable fully polymorphic asymmetric equilibria through saddle–node bifurcations, or to a pair of monomorphic equilibria through transcritical bifurcations. As many as three fully polymorphic stable equilibria can coexist, and jump bifurcations can occur between these equilibria when model parameters are varied.In our Levene model, increasing recombination can act to either increase or decrease the genetic diversity of a population. By generating more hybrid offspring from the mating of purebreds, recombination can act to increase genetic diversity provided the symmetric equilibrium remains stable. But by destabilizing the symmetric equilibrium, recombination can ultimately act to decrease genetic diversity.

The number of equilibria in the diallelic Levene model with multiple demes

The Levene model is the simplest mathematical model to describe the evolution of gene frequencies in spatially subdivided populations. It provides insight into how locally varying selection promotes a population’s genetic diversity. Despite its simplicity, interesting problems have remained unsolved even in the diallelic case.In this paper we answer an open problem by establishing that for two alleles at one locus and J demes, up to 2J−1 polymorphic equilibria may coexist. We first present a proof for the case of stable monomorphisms and then show that the result also holds for protected alleles. These findings allow us to prove that any odd number (up to 2J−1) of equilibria is possible, before we extend the proof to even numbers. We conclude with some numerical results and show that for J>2, the proportion of parameter space affording this maximum is extremely small.

Evolution and polymorphism in the multilocus Levene model with no or weak epistasis

Evolution and the maintenance of polymorphism under the multilocus Levene model with soft selection are studied. The number of loci and alleles, the number of demes, the linkage map, and the degree of dominance are arbitrary, but epistasis is absent or weak. We prove that, without epistasis and under mild, generic conditions, every trajectory converges to a stationary point in linkage equilibrium. Consequently, the equilibrium and stability structure can be determined by investigating the much simpler gene-frequency dynamics on the linkage-equilibrium manifold. For a haploid species an analogous result is shown. For weak epistasis, global convergence to quasi-linkage equilibrium is established. As an application, the maintenance of multilocus polymorphism is explored if the degree of dominance is intermediate at every locus and epistasis is absent or weak. If there are at least two demes, then arbitrarily many multiallelic loci can be maintained polymorphic at a globally asymptotically stable equilibrium. Because this holds for an open set of parameters, such equilibria are structurally stable. If the degree of dominance is not only intermediate but also deme independent, and loci are diallelic, an open set of parameters yielding an internal equilibrium exists only if the number of loci is strictly less than the number of demes. Otherwise, a fully polymorphic equilibrium exists only nongenerically, and if it exists, it consists of a manifold of equilibria. Its dimension is determined. In the absence of genotype-by-environment interaction, however, a manifold of equilibria occurs for an open set of parameters. In this case, the equilibrium structure is not robust to small deviations from no genotype-by-environment interaction. In a quantitative-genetic setting, the assumptions of no epistasis and intermediate dominance are equivalent to assuming that in every deme directional selection acts on a trait that is determined additively, i.e., by nonepistatic loci with dominance. Some of our results are exemplified in this quantitative-genetic context.

What role does natural selection play in speciation?

If distinct biological species are to coexist in sympatry, they must be reproductively isolated and must exploit different limiting resources. A two-niche Levene model is analysed, in which habitat preference and survival depend on underlying additive traits. The population genetics of preference and viability are equivalent. However, there is a linear trade-off between the chances of settling in either niche, whereas viabilities may be constrained arbitrarily. With a convex trade-off, a sexual population evolves a single generalist genotype, whereas with a concave trade-off, disruptive selection favours maximal variance. A pure habitat preference evolves to global linkage equilibrium if mating occurs in a single pool, but remarkably, evolves to pairwise linkage equilibrium within niches if mating is within those niches--independent of the genetics. With a concave trade-off, the population shifts sharply between a unimodal distribution with high gene flow and a bimodal distribution with strong isolation, as the underlying genetic variance increases. However, these alternative states are only simultaneously stable for a narrow parameter range. A sharp threshold is only seen if survival in the 'wrong' niche is low; otherwise, strong isolation is impossible. Gene flow from divergent demes makes speciation much easier in parapatry than in sympatry.

Evolution under the multilocus Levene model without epistasis

Evolution under the multilocus Levene model is investigated. The linkage map is arbitrary, but epistasis is absent. The geometric-mean fitness, w ̃ ( ρ ) , depends only on the vector of gene frequencies, ρ ; it is nondecreasing, and the single-generation change is zero only on the set, Λ , of gametic frequencies at gene-frequency equilibrium. The internal gene-frequency equilibria are the stationary points of w ̃ ( ρ ) . If the equilibrium points ρ ˆ of ρ ( t ) (where t denotes time in generations) are isolated, as is generic, then ρ ( t ) converges as t → ∞ to some ρ ˆ . Generically, ρ ( t ) converges to a local maximum of w ̃ ( ρ ) . Write the vector of gametic frequencies, p , as ( ρ , d ) T , where d represents the vector of linkage disequilibria. If ρ ˆ is a local maximum of w ̃ ( ρ ) , then the equilibrium point ( ρ ˆ , 0 ) T is asymptotically stable. If either there are only two loci or there is no dominance, then d ( t ) → 0 globally as t → ∞ . In the second case, w ̃ ( ρ ) has a unique maximum ρ ˆ and ( ρ ˆ , 0 ) T is globally asymptotically stable. If underdominance and overdominance are excluded, and if at each locus, the degree of dominance is deme independent for every pair of alleles, then the following results also hold. There exists exactly one stable gene-frequency equilibrium (point or manifold), and it is globally attracting. If an internal gene-frequency equilibrium exists, it is globally asymptotically stable. On Λ , (i) the number of demes, Γ , is a generic upper bound on the number of alleles present per locus; and (ii) if every locus is diallelic, generically at most Γ − 1 loci can segregate. Finally, if migration and selection are completely arbitrary except that the latter is uniform (i.e., deme independent), then every uniform selection equilibrium is a migration-selection equilibrium and generically has the same stability as under pure selection.

Polymorphism in the two-locus Levene model with nonepistatic directional selection

For the Levene model with soft selection in two demes, the maintenance of polymorphism at two diallelic loci is studied. Selection is nonepistatic and dominance is intermediate. Thus, there is directional selection in every deme and at every locus. We assume that selection is in opposite directions in the two demes because otherwise no polymorphism is possible. If at one locus there is no dominance, then a complete analysis of the dynamical and equilibrium properties is performed. In particular, a simple necessary and sufficient condition for the existence of an internal equilibrium and sufficient conditions for global asymptotic stability are obtained. These results are extended to deme-independent degree of dominance at one locus. A perturbation analysis establishes structural stability within the full parameter space. In the absence of genotype-environment interaction, which requires deme-independent dominance at both loci, nongeneric equilibrium behavior occurs, and the introduction of arbitrarily small genotype-environment interaction changes the equilibrium structure and may destroy stable polymorphism. The volume of the parameter space for which a (stable) two-locus polymorphism is maintained is computed numerically. It is investigated how this volume depends on the strength of selection and on the dominance relations. If the favorable allele is (partially) dominant in its deme, more than 20% of all parameter combinations lead to a globally asymptotically stable, fully polymorphic equilibrium.

Polymorphism in multiallelic migration–selection models with dominance

The evolution of the gene frequencies at a single multiallelic locus under the joint action of migration and viability selection with dominance is investigated. The monoecious, diploid population is subdivided into finitely many panmictic colonies that exchange adult migrants independently of genotype. Underdominance and overdominance are excluded. If the degree of dominance is deme independent for every pair of alleles, then under the Levene model, the qualitative evolution of the gene frequencies (i.e., the existence and stability of the equilibria) is the same as without dominance. In particular: (i) the number of demes is a generic upper bound on the number of alleles present at equilibrium; (ii) there exists exactly one stable equilibrium, and it is globally attracting; and (iii) if there exists an internal equilibrium, it is globally asymptotically stable. Analytic examples demonstrate that if either the Levene model does not apply or the degree of dominance is deme dependent, then the above results can fail. A complete global analysis of weak migration and weak selection on a recessive allele in two demes is presented.

Evolution under the multiallelic Levene model

The evolution of the multiallelic Levene model is investigated. New sufficient conditions for nonexistence of a completely polymorphic equilibrium and for global loss of an allele and information on which allele(s) will be lost are deduced from some new results on multidimensional recursion relations. In the absence of dominance, a more detailed analysis is presented. Sufficient conditions for global fixation of a particular allele are established. When the number of alleles equals the number of demes, necessary and sufficient conditions for the existence of an isolated, globally asymptotically stable, completely polymorphic equilibrium point are derived, and this equilibrium is explicitly determined. Three examples, one with arbitrarily many alleles and two with three alleles, illustrate the theory.

Ecological Genetics of Mpi and Gpi Polymorphisms in the Acorn Barnacle and the Spatial Scale of Neutral and Non-neutral Variation

Different allozyme genotypes at the mannose phosphate isomerase (Mpi) locus in the northern acorn barnacle (Semibalanus balanoides) show a strong association with distinct intertidal microhabitats. In estuaries along the Maine Coast, the FF homozygote has higher fitness in exposed, high-tide level microhabitats while the SS homozygote has higher fitness under algal cover or at low-tide microhabitats. These patterns are consistent with a Levene (1953) model of balancing selection. In these same samples, polymorphisms at the glucose phosphate isomerase locus (Gpi) and mitochondrial DNA (mtDNA) show no fitness differences among microhabitats, providing intra-genomic controls supporting selection at or near Mpi. Here we report a similar analysis of genotype-by-microhabitat associations at sites in Narragansett Bay, Rhode Island, close to the southern range limit of S. balanoides. Genotype zonation at Mpi between high- and low-tide microhabitats is significantly different between Maine and Narragansett Bay due to opposite zonation patterns for the SF and FF genotypes. Enzyme activity data are consistent with this "reverse" zonation. At Gpi, there is significant microhabitat zonation in Narragansett Bay, while this locus behaves as a neutral marker in Maine. Mt DNA shows no significant microhabitat zonation in either Rhode Island or Maine. The Mpi data suggest that Levene-type selection for alternative genotypes in alternative habitats may operate at scales of both 10's of meters and 100's of kilometers. The Gpi data show how an apparently neutral locus can exhibit non-neutral variation under different environmental conditions. We argue that both Mpi and Gpi provide important genetic variation for adaptation to environmental heterogeneity that is recruited under distinct conditions of stress and carbohydrate substrate availability.

Adaptive maintenance of genetic polymorphism in an intertidal barnacle: habitat- and life-stage-specific survivorship of Mpi genotypes.

In the northern acorn barnacle, Semibalanus balanoides, genotype frequencies of three genetic markers were tracked over time in four types of intertidal habitats. These habitats were selected to represent natural variation in several environmental parameters, specifically the degree of physical stress experienced by barnacles. Frequencies for one allozyme locus (Gpi) and a presumably neutral mtDNA marker were homogeneous among habitats in each temporal sample. Similarly, no temporal stratification in genotype frequencies was evident across the five sampling intervals: from planktonic larvae sampled in March to juveniles collected at the end of June. In contrast to the Gpi and mtDNA loci, Mpi genotypes significantly changed in frequency in two habitats in the high intertidal zone. On exposed substrate, the Mpi-FF homozygote increased in frequency, whereas the alternative homozygote, Mpi-SS, significantly decreased in frequency. Barnacles that were protected from environmental stress at high intertidal heights by the Ascophyllum nodosum algal canopy demonstrated the opposite pattern. In both habitats, the change in frequency of the heterozygote was intermediate to that of the homozygous genotypes. Furthermore, these patterns of genotype-by-environment association reflected a pulse of genotype-specific mortality that occurred over a two-week interval subsequent to metamorphosis from the larval to the adult form. These data indicate that each Mpi homozygote is the highest fitness genotype in some portion of the intertidal environment. Using the Levene (1953) model to evaluate the spatial variation in genotypic fitness, the stable maintenance of the Mpi polymorphism is predicted under certain subsets of conditions. Environmental heterogeneity in the intertidal zone translates to spatial variation in selection pressures, which may result in the active maintenance of the Mpi polymorphism in this species.

Patterns of Multiallelic Polymorphism Maintained by Migration and Selection

Evolution at a multiallelic locus under the joint action of migration and viability selection is investigated. Generations are discrete and nonoverlapping. The monoecious, diploid population is subdivided into finitely many panmictic colonies that exchange adult migrants independently of genotype. The forward migration matrix is arbitrary, but time independent and ergodic (i.e., irreducible and aperiodic). Several examples of globally attracting multiallelic equilibria are presented. Migration can cause global fixation even if, without migration, there is a globally attracting multiallelic equilibrium in every colony. Migration can also cause the global fixation of an allele that, without migration, is eliminated in every colony. Without dominance, generically, the number of alleles present at equilibrium cannot exceed the number of colonies. Some general properties and examples of the Levene model are studied in detail. If in each colony there is either no dominance or, without migration, a globally attracting internal equilibrium, then there exists a globally attracting equilibrium with migration. Therefore, if an internal equilibrium exists, it is the global attractor.

ADAPTIVE MAINTENANCE OF GENETIC POLYMORPHISM IN AN INTERTIDAL BARNACLE: HABITAT- AND LIFE-STAGE-SPECIFIC SURVIVORSHIP OF MPI GENOTYPES

In the northern acorn barnacle, Semibalanus balanoides, genotype frequencies of three genetic markers were tracked over time in four types of intertidal habitats. These habitats were selected to represent natural variation in several environmental parameters, specifically the degree of physical stress experienced by barnacles. Frequencies for one allozyme locus (Gpi) and a presumably neutral mtDNA marker were homogeneous among habitats in each temporal sample. Similarly, no temporal stratification in genotype frequencies was evident across the five sampling intervals: from planktonic larvae sampled in March to juveniles collected at the end of June. In contrast to the Gpi and mtDNA loci, Mpi genotypes significantly changed in frequency in two habitats in the high intertidal zone. On exposed substrate, the Mpi-FF homozygote increased in frequency, whereas the alternative homozygote, Mpi-SS, significantly decreased in frequency. Barnacles that were protected from environmental stress at high intertidal heights by the Ascophyllum nodosum algal canopy demonstrated the opposite pattern. In both habitats, the change in frequency of the heterozygote was intermediate to that of the homozygous genotypes. Furthermore, these patterns of genotype-by-environment association reflected a pulse of genotype-specific mortality that occurred over a two-week interval subsequent to metamorphosis from the larval to the adult form. These data indicate that each Mpi homozygote is the highest fitness genotype in some portion of the intertidal environment. Using the Levene (1953) model to evaluate the spatial variation in genotypic fitness, the stable maintenance of the Mpi polymorphism is predicted under certain subsets of conditions. Environmental heterogeneity in the intertidal zone translates to spatial variation in selection pressures, which may result in the active maintenance of the Mpi polymorphism in this species.Corresponding Editor: R. Burton

COMPETITION AND GENOTYPE-BY-ENVIRONMENT INTERACTION IN NATURAL BREEDING SUBSTRATES OF DROSOPHILA.

Although empirical studies frequently suggest that genotype-by-environment (G X E) interaction can maintain genetic variation, very few data are available to test for the specific conditions necessary for the existence of a protected polymorphism (i.e., the property of persistence of an allele even when initially rare). Drosophila species live in patchy environments and their local population structure may be characterized to some extent by Levene's migration pattern, namely by a single pool of individuals that presumably mate at random and breed on discrete and ephemeral resources. We present here a field experiment that links Drosophila ecology and population genetics, which used the alcohol dehydrogenase (Adh) and α-glycerophosphate dehydrogenase (αGpdh) polymorphic loci in D. melanogaster flies raised from Opuntia ficus-indica fruits (prickly pears). The results show that there is density-dependent mortality in those fruits with a relatively high number of larvae (i.e., selection is "soft") and suggest that there is differential viability for αGpdh genotypes. Additionally, a pattern of G X E interaction for fitness values, which is fully compatible with the theoretical conditions required for the existence of a protected polymorphism, was found after weighting the fitness estimates by the relative contribution that each fruit makes to the total adult population. The strong association between AdhS and αGpdhF alleles suggests that the occurrence of the common cosmopolitan inversion In(2L)t in the population might be responsible for the negative frequency-dependent selection predicted by Levene's model when genetic variation persists in heterogeneous environments.

INTERTIDAL MICROHABITAT AND SELECTION AT MPI: INTERLOCUS CONTRASTS IN THE NORTHERN ACORN BARNACLE, SEMIBALANUS BALANOIDES.

Barnacles were sampled from various microhabitats in the rocky intertidal at multiple sites in two years. At sites in which there were large differences among microhabitats in temperature profiles, Mpi genotype frequencies were consistently and significantly different. Genotype frequencies for another allozyme locus (Gpi) as well as a DNA marker shown to be neutral (the mtDNA control region) were statistically homogeneous among thermal microhabitats at all sites in both years. The data indicate that temperature and/or desiccation mediated selection is operating at Mpi or a linked locus and that Mpi genotypes experience differential mortality in the various habitat types. If the relative fitness of genotypes is dependent on habitat type, the Mpi polymorphism may be actively maintained by a Levene model of balancing selection (Levene 1953). Because barnacle larvae are produced in abundance each year and spend five to eight weeks dispersing in the water column, there is little opportunity for the accumulation of adaptive divergence over the environmental grain size relevant in intertidal habitats. The Mpi polymorphism may be an important component of a suite of traits involved in the adaptation of barnacles to heterogeneous environments. Due to the relatively high concentration of mannose in a variety of algal groups, the metabolism of mannose may substantially affect individual performance and fitness in this and other species that feed on algae and phytoplankton. Because the Mpi locus is one of the most strongly polymorphic in marine organisms, these findings may be relevant for a diversity of other such species.

Epistatic Selection in a Multi-locus Levene Model and Implications for Linkage Disequilibrium

We analyze a multiple-locus extension of the Levene (1953) model of population subdivision. We show that stable or quasistable linkage disequilibrium between two selected loci can be maintained even with free recombination, provided that there is a strong enough epistatic interaction. We then consider the dynamics of a third neutral locus and show that its approach to linkage equilibrium depends on the recombination ratesandthe selection intensities. There is an embedding or hitchhiking effect that extends the time during which a neutral locus which is closely linked to one of the selected loci remains in disequilibrium with both selected loci. Therefore, strong disequilibrium between two loci does not necessarily indicate that those loci are themselves selected, but it does indicate that there is strong selection acting at least on nearby loci. This property implies a warning that screening for linkage disequilibrium as a tool to identify functionally important sites in a genome can be misleading.

Hypoxia-ischaemia model in the 7-day-old rat: possibilities and shortcomings.

The Levene model in 7-day-old rats is the most often used model of hypoxia-ischaemia (HI) in immature animals. The rat central nervous system is immature at birth and corresponds neurodevelopmentally to the term human infant during the second postnatal week. The Levene model of HI differs from clinical asphyxia with respect to the unilateral distribution of brain injury and lack of multi-organ dysfunction. Furthermore, it does not allow cardiovascular monitoring or repeated blood sampling. On the other hand, the progressive nature of HI bears many similarities to birth asphyxia with regard to blood flow changes and cellular metabolic derangements. The model is well characterized, easy to carry out and the low cost allows inclusion of a sufficient number of animals for dose-response evaluation of neuroprotective agents. In addition, it provides the unique opportunity of long-term evaluation of neuropathological and functional outcome.

Selection and Population Regulation with Habitat Variation

Sufficient conditions for protected polymorphism at an autosomal locus in a diploid organism are developed for a Levene model with explicit modeling of the population dynamics. Levene's condition for protected polymorphism, formulated in terms of weighted harmonic means of relative genotypic fitnesses, is only strictly valid for components of selection that are not immediately relevant for the regulation of the population size in the subpopulations. Global regulation of the population size produces Dempster's condition for protected polymorphism in terms of weighted arithmetic means of absolute genotypic fitnesses. The general condition for protection of an allele may be written in the same form as Levene's condition, but then the weights of the subpopulations depend on the fitnesses in the various habitats. The model allows for marginal habitats that cannot sustain a population if left isolated.

Density-Regulated Selection in a Heterogeneous Environment

A model is developed for density-regulated selection in a heterogeneous environment, analogous to Levene's model for selection in a subdivided environment. The population is polymorphic for two alleles at an autosomal locus. Before selection the three genotypes are in Hardy-Weinberg frequencies, and a constant fraction of newborn offspring go to habitat k to live. Genotypes differ in their selective values because of genotypic and habitat differences in intrinsic rates of increase and carrying capacities, and selective values are parameterized in terms of these ecological fitness parameters so that the growth of the whole population is logistic. All selection occurs in the habitats, and at the end of each generation adults emerge from the habitats and join a mating pool to mate at random. In the Levene model, individuals in the mating pool from each habitat make a constant contribution to the gametic pool from which individuals of the next generation develop. In our model, however, the gametic contribution from a habitat is proportional to the number of organisms remaining in the habitat after selection has taken place, and thus to the average selective value of individuals from the habitat. As a consequence, the gametic contributions change with allele frequency. Sufficient conditions for a protected polymorphism under this more realistic assumption are derived, and remarkably enough take a form analogous to Levene's, with the same increased opportunity for genetic polymorphism which is the major result of his model. A protected polymorphism occurs provided that a harmonic mean over habitats of the adjusted carrying capacities of each homozygote are less than a corresponding mean for the heterozygote; the adjusted carrying capacity of a genotype in a habitat is just an extension to the population level of the carrying capacity for a single habitat. The population's carrying capacity, a weighted harmonic mean of the adjusted carrying capacities of the genotypes, is maximized near population size equilibrium. Sufficient conditions for a protected polymorphism under a more general form of density-regulated selection, with the genotypic fitnesses any smooth, strictly decreasing function of population size, are also outlined.

Migration-selection polymorphism in dioecious populations

Conditions are derived for a protected polymorphism in a dioecious population subdivided into an arbitrary number of demes which exchange migrants. Generations are discrete and nonoverlapping; mutation and random drift are neglected. The analysis is restricted to a diallelic autosomal locus. In contrast to the monoecious case, the protection criteria depend on the order of migration and selection; they become identical for adult and juvenile migration if both the male and female backward migration matrices are symmetric, or the migration or selection patterns in the two sexes are the same. The protection conditions are presented explicitly for the Levene model. A recessive allele is protected in a panmictic dioecious population if the unweighted average of the recessive-to-dominant fitness ratios in the two sexes exceeds unity.